# Chain complex

In

homological algebra, a

**chain complex** is a sequence of abelian groups or modules

*A*_{0},

*A*_{1},

*A*_{2}... connected by homomorphisms

*d*_{n} :

*A*_{n} ` -> ` *A*_{n-1}, such that the composition of any two consecutive maps is zero:

*d*_{n} o

*d*_{n+1} = 0 for all

*n*.
Chain complexes are mainly used to define

homology and cohomology.

*Examples from topology, group theory...*